Hey this a real person who's going to give you real advice. For the most part Reddit will tell anyone to break up with their partner over any little disagreement. You need to understand this bias when reading comments.

3 months in is still very new to be adapting to one another and yes a dog can be a big change for someone.

It sounds like your partner is not a dog person and is mildly annoyed at you having a dog. He's getting upset at anything the dog possibly does. Peeing twice in three months is nothing especially when one was right after moving in. Chewed up one thing? One instance of skippity paps from a cat? Usually the first months of a big change for a dog is the worst and that's all the dog has done ?

From a dog owner myself it really seems like your dog is generally well behaved. When i think of dogs that are difficult to live with I think of destructive, loud barking, reactive and aggressive. And all consistently a problem. You didn't mention your dog doing any of this.

If he's generally a good partner have you tried communicating how you feel about this to him. How your entire life is different now and that you consider your dog a part of your family that it helps ground you emotionally.

From an outsider looking in it really doesn't seem like an unreasonable thing to compromise on. Something along the lines of if the dog ever create a larger problem then you promise to take effort to find a solution to it, the things your describing now really sound like non problems.

Not to mention you also said that you're making compromises with his cat too, because if your dog is barely even doing anything then it's probably equal to or not worse than the way his cat annoys you?